Group #21426144 — subgroups

Subgroups — #21426144

Order 7552.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14273673	C₂	1	1	yes	maximal, characteristic
2	#16837780	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic
4	Dic₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#16837207	C₂	1	1	yes	maximal, characteristic
6	#16837206	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
9	C₂×#11745672	—	2	1	no	—
10	#14273725	—	2	1	no	—
11	Q₈×Dic₅₉	—	2	1	no	—
12	#14273673	—	2	2	no	—
13	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
14	#14273661	C₂²	1	1	yes	characteristic
15	#14273651	—	2	1	no	—
16	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
19	Q₃₂×C₅₉	—	2	2	no	—
20	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₅₉	—	2	1	no	—
22	C₂×Q₈×C₅₉	—	2	1	no	—
23	C₂×Dic₁₁₈	—	2	1	no	—
24	Q₁₆×C₅₉	—	2	1	no	—
25	Q₁₆×C₅₉	—	2	1	no	—
26	Q₁₆×C₅₉	—	2	1	no	—
27	Q₁₆×C₅₉	D₄	1	2	yes	—
28	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	#11745672	—	4	2	no	—
30	C₄⋊Dic₅₉	—	2	1	no	—
31	C₄⋊Dic₅₉	—	4	1	no	—
32	Dic₅₉⋊C₄	—	4	1	no	—
33	Dic₅₉⋊C₄	—	4	1	no	—
34	C₄×Dic₅₉	—	4	1	no	—
35	C₄×Dic₅₉	D₄	1	1	yes	characteristic
36	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
37	Dic₂₃₆	—	2	1	no	—
38	Dic₂₃₆	—	2	1	no	—
39	C₁₆×C₅₉	—	2	1	no	—
40	C₅₉⋊C₁₆	—	2	1	no	—
41	Q₈×C₅₉	—	2	2	no	—
42	Q₈×C₅₉	—	4	1	no	—
43	Q₈×C₅₉	—	4	1	no	—
44	Q₈×C₅₉	—	4	1	no	—
45	C₂×C₄×C₅₉	—	4	1	no	—
46	C₂×C₄×C₅₉	—	4	1	no	—
47	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₅₉	—	4	1	no	—
49	C₂×Dic₅₉	—	4	1	no	—
50	C₂×Dic₅₉	D₈	1	1	yes	characteristic
51	C₂×Dic₅₉	D₈	1	1	yes	characteristic
52	Dic₁₁₈	—	4	1	no	—
53	Dic₁₁₈	—	4	1	no	—
54	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₈×C₅₉	#41	1	1	yes	characteristic
56	C₅₉⋊C₈	—	2	1	no	—
57	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
58	C₄×C₅₉	—	4	2	no	—
59	C₄×C₅₉	—	8	1	no	—
60	C₄×C₅₉	#121	1	1	yes	characteristic
61	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₅₉	—	2	1	no	—
63	Dic₅₉	—	8	1	no	—
64	Dic₅₉	—	8	1	no	—
65	Dic₅₉	Q₃₂	1	2	yes	complemented
66	#2209	—	59	1	no	maximal, Sylow
67	C₂×C₅₉	#459	1	1	yes	characteristic
68	C₂×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
69	C₂×C₅₉	#510	1	1	yes	characteristic
70	C₂×Q₃₂	—	59	1	no	—
71	C₂×Q₃₂	D₅₉	1	1	yes	characteristic
72	#494	—	59	1	no	—
73	C₄×Q₁₆	—	59	1	no	—
74	C₄⋊C₁₆	—	59	1	no	—
75	#358	—	59	1	no	—
76	#358	—	59	1	no	—
77	C₅₉	#2209	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₁₆	—	59	1	no	—
79	C₂×Q₁₆	—	118	1	no	—
80	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
81	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
82	C₄⋊Q₈	—	118	1	no	—
83	C₄×Q₈	—	118	1	no	—
84	Q₃₂	—	2	2	no	—
85	Q₃₂	—	118	2	no	—
86	C₂×C₁₆	—	59	1	no	—
87	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
88	#107	—	59	1	no	—
89	#103	—	118	1	no	—
90	C₄×C₈	—	59	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	118	1	no	—
94	Q₁₆	—	2	1	no	—
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	2	1	no	—
97	Q₁₆	—	118	1	no	—
98	Q₁₆	—	118	1	no	—
99	Q₁₆	—	236	2	no	—
100	Q₁₆	C₂²⋊D₅₉	1	2	yes	—
101	C₂×C₈	—	59	1	no	—
102	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
103	C₄⋊C₄	—	118	1	no	—
104	C₄⋊C₄	—	236	1	no	—
105	C₄⋊C₄	—	236	1	no	—
106	C₄⋊C₄	—	236	1	no	—
107	C₄²	—	59	1	no	—
108	C₄²	—	236	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	118	1	no	—
111	Q₈	—	2	2	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	4	1	no	—
114	Q₈	—	4	1	no	—
115	Q₈	—	236	1	no	—
116	Q₈	—	236	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	59	1	no	—
120	C₂×C₄	—	59	1	no	—
121	C₂×C₄	—	236	1	no	—
122	C₂×C₄	—	236	1	no	—
123	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
124	C₈	—	118	1	no	—
125	C₈	#11745687	1	1	yes	characteristic
126	C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
127	C₂²	D₈×D₅₉	1	1	yes	characteristic
128	C₄	—	4	2	no	—
129	C₄	—	8	1	no	—
130	C₄	—	59	2	no	—
131	C₄	—	118	1	no	—
132	C₄	—	472	1	no	—
133	C₄	—	472	1	no	—
134	C₄	#14273699	1	1	yes	characteristic
135	C₄	#14273751	1	1	yes	characteristic
136	C₂	#16837595	1	1	yes	characteristic
137	C₂	Q₃₂×D₅₉	1	1	yes	characteristic
138	C₂	#16837844	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273673

C₂

yes

maximal, characteristic

#16837780

C₂

yes

maximal, characteristic

Q₁₆×Dic₅₉

C₂

yes

maximal, characteristic

Dic₅₉⋊C₁₆

C₂

yes

maximal, characteristic

#16837207

C₂

yes

maximal, characteristic

#16837206

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₅₉

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₂×#11745672

—

#14273725

—

Q₈×Dic₅₉

—

#14273673

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273651

—

C₈×Dic₅₉

C₂²

yes

characteristic

C₂×Q₁₆×C₅₉

C₂²

yes

characteristic

C₂×Q₁₆×C₅₉

C₂²

yes

characteristic

Q₃₂×C₅₉

—

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Dic₁₁₈

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

#11745672

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Dic₂₃₆

—

Dic₂₃₆

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

characteristic

C₂×Dic₅₉

D₈

yes

characteristic

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

Q₃₂

yes

—

C₂×C₅₉

yes

C₂×C₅₉

C₂×Q₃₂

yes

characteristic

C₂×C₅₉

#510

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

D₅₉

yes

characteristic

#494

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

#358

—

#358

—

C₅₉

#2209

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

118

—

C₂×Q₁₆

C₂×D₅₉

yes

characteristic

C₂×Q₁₆

C₂×D₅₉

yes

characteristic

C₄⋊Q₈

—

118

—

C₄×Q₈

—

118

—

Q₃₂

—

Q₃₂

—

118

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

#107

—

#103

—

118

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

118

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

118

—

Q₁₆

—

118

—

Q₁₆

—

236

—

100

Q₁₆

C₂²⋊D₅₉

yes

—

101

C₂×C₈

—

102

C₂×C₈

C₂²×D₅₉

yes

characteristic

103

C₄⋊C₄

—

118

—

104

C₄⋊C₄

—

236

—

105

C₄⋊C₄

—

236

—

106

C₄⋊C₄

—

236

—

107

C₄²

—

108

C₄²

—

236

—

109

C₁₆

—

110

C₁₆

—

118

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

236

—

116

Q₈

—

236

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

236

—

122

C₂×C₄

—

236

—

123

C₂×C₄

D₄×D₅₉

yes

characteristic

124

C₈

—

118

—

125

C₈

#11745687

yes

characteristic

126

C₈

C₂×C₂²⋊D₅₉

yes

characteristic

127

C₂²

D₈×D₅₉

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

118

—

132

C₄

—

472

—

133

C₄

—

472

—

134

C₄

#14273699

yes

characteristic

135

C₄

#14273751

yes

characteristic

136

C₂

#16837595

yes

characteristic

137

C₂

Q₃₂×D₅₉

yes

characteristic

138

C₂

#16837844

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₂×Q₃₂×C₅₉	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	C₂×Q₃₂	D₅₉
hypocenter	C₅₉	#2209
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₂×Q₃₂×C₅₉

C₂

socle

C₂²×C₅₉

C₂×D₈

hypercenter

C₂×Q₃₂

D₅₉

hypocenter

C₅₉

#2209

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
65. Dic₅₉	Q₃₂	semidirect
77. C₅₉	#2209	semidirect

H ≅ G/N

type

65. Dic₅₉

Q₃₂

semidirect

77. C₅₉

#2209

semidirect

Subgroups — #21426144

Conjugacy classes of subgroups

Complements